This study proposes a bootstrap-based method for uncertainty quantification in two important statistical scenarios. First, we approximate the sampling distribution of empirical barycenters under the Bures--Wasserstein metric using a reweighted estimator. Our theoretical results guarantee the accuracy of this approximation and enable the construction of data-driven confidence sets. The methodology is validated through experiments on graph-structured data, including stochastic block models and brain connectomes. Additionally, we compare bootstrap-based confidence sets with the asymptotic confidence sets obtained in arXiv:1901.00226v2, evaluating both their statistical performance and computational complexity. Second, we investigate the generalized bootstrap framework for $M$-estimators without requiring a specific resampling scheme, thus covering both weighted and resampling methods under mild conditions. Both contributions rely on a novel Gaussian approximation result for $M$-estimators.

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